You can apply the same logic to a specific license plate number seen near a murder meaning that the car's owner was almost surely the murderer. That isn't a high enough standard of evidence as any specific event is unlikely in hindsight.

A pandemic was likely to start somewhere densely populated and first be detected in a city centre. Somewhere densely populated was likely to have a specialised biological research lab. That it was BSL4 instead of BSL3 is evidence but it isn't particularly compelling.

More to the point; it doesn't really make much of a difference whether it came from a Chinese lab or not. Horrible pandemics have happened before and will happen again.

In the murder scenario, we would have something like

  P(individual is guilty) = 1 / 7 billion
  P(individual's car on same block | individual is guilty) = 1
  P(individual's car on same block | individual is innocent) = 50 / 7 billion
  P(individual is guilty | individual's car on same block) = 1 / 50 (from Bayes' rule)

You might disagree with the priors, but I'm pretty sure the argument is sound.

The soundness of the argument isn't the issue. The problem is the world model where P(Virus starts in Wuhan) = 1.

That probability is only 1 in hindsight, where the probability of everything is 1 because it has already happened. In foresight the probability was not 1. And whatever city the virus started in was going to have a BSL-2+ facility studying infectious diseases. I'm probably sitting within 2-4km of a BSL-3 facility studying infectious diseases as I type this, because I am in a city.

The fast that WIV is BSL-4 is significant, but it isn't going to get you to 97%. Even 75% is high. Particularly if WIV is located where it is because there are good local sources of coronaviruses to study.

I don't quite follow your point. My model assumed implicit knowledge that the virus originated somewhere in China, but not in Wuhan. Maybe I should have made that more explicit.

We could change the model to not have any implicit assumption about location. Then instead of dividing by the population of China, we'd divide by the world population. The result should be the same, assuming we appropriately adjust our prior probability P(virus from WIV) to account for the fact that there are more BSL-4 facilities worldwide than in China.

Well, I'm not qualified to explain this so I'm probably wrong in the detail, but a sketch of the thinking:

P(Came from WIV|Global Epidemic starts in Wuhan) ~= 97%

However, P(Started Near a BSL Rated facility|Global Epidemic) is also high (not going to estimate it).

But you have assumed that P(Global Epidemic starts in Wuhan) = 1; which is not true. P(Global Epidemic starts in Wuhan) is probably quite low and they got unlucky because it had to start somewhere. In this context, that assumption is tantamount to flipping a coin then saying "P(Heads) = 1". It isn't, P(Head) = 0.5 but it does happen sometimes.

If we assume P(Global Epidemic starts in Wuhan) = 1 (because it did happen) then P(Came from WIV) is either 0 or 1, because it either did or didn't happen in the same way that the virus started in Wuhan.

Hm, you're making sense otherwise but I just don't see where I assumed P(Global Epidemic starts in Wuhan) = 1 -- could you elaborate on that?

Probably not; no.

The best I can come up with is that probability is very sensitive to what you assume is happening to generate results. In this case, you're assuming that the WIV in particular is important which I suspect links the pandemic implicitly to Wuhan. That isn't a math error, but it is not going to give you a useful result.

This isn't an error of logic so much as an error in scenario construction - it is ignoring possibility that the virus appeared near a random BSL rated facility, which is probably very high since a new virus would likely be detected first in dense urban centers where that sort of infrastructure tends to be. If the virus is likely to appear near a BSL facility, then the fact it appeared near a specific BSL facility shouldn't be leading you to the sort of answer you got.

maybe someone can correct me, but AFAIK, BSL4 is required for handling SARS-related coronavirus samples, at least in the US. this would make the distinction between BSL3 and BSL4 pretty significant. while there probably are BSL3 labs in most major cities in the world, there are only two BSL4 facilities in china and WIV is one of them.

Now this is an interesting abuse of Bayes' theorem, because there's an extremely similar conspiracy theory going around in China of this form: it's known that CIA agents specifically visited Wuhan before the virus broke out, and the CIA claims to have known about this disease before anybody in China did. So...

    P(virus from CIA) = 0.01
    P(outbreak in 14km radius | virus from CIA) = 1
    P(outbreak in 14km radius | virus not from CIA) = 1/3,000

Then Bayes' rule gives P(virus from CIA | outbreak in 14km radius) ~= 97%.

But wait a second! How can there both be a 97% chance that the CIA did it, and a 97% chance that WIV made it? The reason is that both of these calculations are total bullshit.

What do you mean by P(virus from CIA)? Do you mean the probability that the CIA planted the virus in the WIV? If so, then

> How can there both be a 97% chance that the CIA did it, and a 97% chance that WIV made it?

There's no contradiction there. If the CIA planted the virus in the WIV, then it's simultaneously true that the CIA planted the virus and that the WIV was the source of the outbreak.

Why not both?

Using your exact logic three times:

- there's a 97% chance the CIA planted it in the seafood market

- there's a 97% chance the CIA planted it in the WIV

- there's a 97% chance the WIV made it

Now it's obvious that something is going wrong here. There is no conceivable way these probability assignments can make sense. But your math was formally correct! What a mystery.

I threw out 1% as a seemingly reasonable prior probability that the outbreak came from the WIV, given an implicit assumption that the virus originated in China. Obviously I didn't put much thought into the exact number, but 1% seems within the realm of reason to me, given that the WIV is the only BSL-4 facility in China.

It doesn't make sense to apply the same prior to "the CIA planted COVID-19 in the Huanan Seafood Wholesale Market", because there are many such markets in China, and only one BSL-4 facility. You might think 1% is a reasonable prior for "the CIA planted COVID-19 in some market in China", but then it wouldn't make sense to adjust the probability based on the outbreak location as I did.

What you're doing is like starting with a prior probability of 10% that a random individual lives in Maui, and 10% that a random individual lives in Kawaii. Then you learn that an individual lives in Hawaii, and you try to adjust both probabilities based on Hawaii's population in relation to the world population. Bayes' theorem gives ~99.8% each for P(Maui | Hawii) and P(Kawaii | Hawaii). The results are obviously impossible, but the argument wasn't structurally wrong, you just started with bad priors. You should have started with P(Kawaii) ~= 0.001%, giving P(Kawaii | Hawaii) ~= 5%, which is correct.

> It doesn't make sense to apply the same prior to "the CIA planted COVID-19 in the Huanan Seafood Wholesale Market", because there are many such markets in China, and only one BSL-4 facility.

I didn't make an error. The way the conspiracy theory goes in China is that the Military World Games [0] were held in Wuhan just before the outbreak. In other words, they knew the US military was in Wuhan specifically (in fact, right next to the seafood market), just like you know that WIV is in Wuhan specifically. The two conspiracy theories really are equally credible, by which I mean they're both bullshit.

The reason is because of this assumption you made:

    P(my favorite conspiracy theory) = 0.01
    P(outbreak in 14km radius | my favorite conspiracy theory is false) = 1/3,000

Because 1/3000 is so much lower than 1/100, you're essentially assuming from the start that your favorite conspiracy theory (insert "WIV" or "CIA" or whatever else here) is a much better explanation than all other possible explanations combined, because all of them combined can only total up to a likelihood of 1/3000. In other words, you rigged the game to support your conspiracy theory from the start. I was trying to point out how similar rigging can be used to justify any conspiracy theory.

0: https://en.wikipedia.org/wiki/2019_Military_World_Games

Yes, I assumed that if the virus is not from the WIV, then all areas of China (weighted by population) would be equiprobable candidates for the outbreak location. Of course this is an imperfect model, but I think it's a reasonable one.

This alternate conspiracy seems way more farfetched to me (why would US intelligence need such any such event to get an agent into China?), but if someone believed that it should be given a non-negligible prior, then they should revise my assumption that

> P(outbreak in 14km radius | virus not from WIV) = 1/3,000

I.e. their revised value should be at least as large as their prior for the alternate conspiracy.

What part of the argument are you disputing exactly? Are you claiming that applying Bayes' theorem in this way is somehow invalid, or are you just taking issue with the assumption quoted above?

Thanks for using Bayes theorem. Although it does not provide evidence, it should guide us to where we should look for evidence.

I am also very surprised at people throwing probabilistic equivalence between foreign-agent conspiracy, and an accident in a lab. It seems that the latter is so much more probable.

Your prior is also exactly what you're trying to calculate, though. I don't think you can just try different numbers picked from a hat for that value.

Also, it isn't accounting for lots of other highly relevant, potentially confounding variables.

And those assumptions make no sense to me at all. Can you write them out much more explicitly?

I think it's totally plausible it was a lab escape, but this analysis seems extremely oversimplified and naive, and I'd even say a misuse and abuse of Bayes' theorem. A full Bayesian analysis is a good idea, though. I'd like to see a much more robust and detailed one from someone.

> Your prior is also exactly what you're trying to calculate, though.

Maybe I should have been more explicit, but they're not the same. The prior here is the probability that the virus originated from WIV, knowing only that the virus originated in China, and not accounting for the location of the outbreak. I'm trying to calculate an adjusted probability given the outbreak location.

> I don't think you can just try different numbers picked from a hat for that value.

My goal was not to convince anyone that a prior of 1% or 0.1% is appropriate, but rather to suggest a framework in which people can insert their own priors.

And sure, it's a simplified analysis, but it's better than just asserting whatever our intuitions say about coincidences.

> The prior here is the probability that the virus originated from WIV, knowing only that the virus originated in China, and not accounting for the location of the outbreak.

Makes more sense, but as you say, there's really no way of knowing what number is appropriate there. I'm not a statistics expert, but I'd say it's inextricably tied up with what you're trying to calculate the probability of in the first place. The whole point is we don't have a prior for WIV being likely to have a virus escape the lab, or a coronavirus, or this specific virus.

I think you'd need to break it down into priors like: probability any Chinese lab has any virus escape, probability this particular lab has any virus escape, probably this particular lab had any virus escape around this time frame, probability this particular lab had a sample of SARS-CoV-2, and a lot more things.

>And sure, it's a simplified analysis, but it's better than just asserting whatever our intuitions say about coincidences.

It certainly can be much better in general, and I'm a huge fan of Bayesianism, but it can also sometimes provide a very misleading impression to others. I'm somewhat tempted to try to do a deep dive into a Bayesian analysis of this, but I'm not really qualified, and it would be a ton of work, and there would be some priors that would be nearly impossible to know (number and density of animals of certain species in the area, what pathogens they may have, what animals and viruses the lab may have collected or been interested in collecting, what and how many animals are hunted in the area, what and how many hunted animals are sold at local wet markets, how such animals are handled or prepared, etc.)

I think to assess this with any serious confidence, we're probably going to need something like leaked/stolen documents or a whistleblower.

The Wuhan Center for Disease Control, which routinely sent viral samples to WIV, is only 300 meters from the wet market. The 300 meter radius circle surrounding the Wuhan Center for Disease Control is 0.33 square miles. China is 3,705,009 square miles. The chances of this happening so close to the Wuhan Center for Disease Control is therefore 1 in 11.2 million.